Let $\{A_1,A_2,...,A_{n^2-1}\}$ be one basis for the subspace $M$, $M \le M_n$, where $M$ is a subspace of all matrices whose trace is $0$. Determine the matrice of linear tranformation $\operatorname{tr}:M_n \rightarrow \Bbb F$ relative to the basis $\{A_1, A_2,...,A_{n^2-1}, 6I\}$ of $M_n$ and basis $\{3\}$ of $\Bbb F$.
My attempt:
The trace of all of the matrices $A_1, A_2,...,A_{n^2-1}$ is $0$, and $\operatorname{tr}(6I)=6n=3\cdot 2n$ which means that the matrice of the transformation is:
$$ \begin{bmatrix} 0 & 0 & \cdots& 0 &2n \\ \end{bmatrix} $$
Is this correct and, if it is, is this a good proof or do I need to explain something further?